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Hardy spaces of vector-valued functions: duality


Author: Oscar Blasco
Journal: Trans. Amer. Math. Soc. 308 (1988), 495-507
MSC: Primary 46E40; Secondary 28B05, 42B30, 46E30
DOI: https://doi.org/10.1090/S0002-9947-1988-0951618-8
MathSciNet review: 951618
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Abstract: We prove here that the Hardy space of $ B$-valued functions $ {H^1}(B)$ defined by using the conjugate function and the one defined in terms of $ B$-valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on $ B$. We also characterize the dual space of both spaces, the first one by using $ {B^{\ast}}$-valued distributions and the second one in terms of a new space of vector-valued measures, denoted $ \mathcal{B}\mathcal{M}\mathcal{O}({B^{\ast}})$, which coincides with the classical $ \operatorname{BMO} ({B^{\ast}})$ of functions when $ {B^{\ast}}$ has the RNP.


References [Enhancements On Off] (What's this?)

  • [1] O. Blasco, On the dual of $ H_B^1$, Colloq. Math. (to appear). MR 978922 (89m:46068)
  • [2] -, Positive $ p$-summing operators on $ {L_p}$-spaces, Proc. Amer. Math. Soc. 100 (1987), 275-280. MR 884466 (88c:47033)
  • [3] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), 163-168. MR 727340 (85a:46011)
  • [4] -, Vector valued singular integrals and $ {H^1} - \operatorname{BMO} $ duality, Probability Theory and Harmonic Analysis, J. A. Chao and W. A. Woyczynski, editors, Marcel Dekker, New York, 1986, pp. 1-19.
  • [5] A. V. Bukhvalov, Duals of spaces of vector-valued analytic functions and duality of functions generated by these spaces, LOMI 92 (1979), 30-50. MR 566740 (81b:46052)
  • [6] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Conf. Harmonic Analysis in honor of A. Zygmund, Wadsworth, Belmont, Calif., 1982, pp. 270-286. MR 730072 (85i:42020)
  • [7] -, Martingales and Fourier analysis in Banach spaces, Lecture Notes in Math., vol. 1206, Springer-Verlag, Berlin, 1986. MR 864712 (88c:42017)
  • [8] D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal function characterization of the class $ {H^p}$, Trans. Amer. Math. Soc. 157 (1971), 137-153. MR 0274767 (43:527)
  • [9] S.-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and $ {H^p}$-theory in product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 1-43. MR 766959 (86g:42038)
  • [10] R. R. Coifman, A real variable characterization of $ {H^p}$, Studia Math. 51 (1974), 269-274. MR 0358318 (50:10784)
  • [11] R. R. Coifman and G. Weiss, Extension of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569-645. MR 0447954 (56:6264)
  • [12] J. Diestel and J. J. Uhl, Vector measures, Math. Surveys, no. 15, Amer. Math. Soc., Providence, R.I., 1977. MR 0453964 (56:12216)
  • [13] N. Dinculeanu, Linear operators on $ {L^p}$-spaces. Vector and operator valued measures and applications, Proc. Sympos. Snowbird Resort (Alta, Utah), Academic Press, New York, 1972, pp. 109-124. MR 0341066 (49:5816)
  • [14] P. L. Duren, Theory of $ {H^p}$-spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [15] C. Fefferman, Characterization of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587-588. MR 0280994 (43:6713)
  • [16] C. Fefferman and E. M. Stein, $ {H^p}$-spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [17] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985. MR 807149 (87d:42023)
  • [18] W. Hensgen, Hardy-Raume vektorwertiger Funktionen, Thesis, München 1986.
  • [19] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 0131498 (24:A1348)
  • [20] J. L. Journé, Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, Lecture Notes in Math., vol. 994, Springer-Verlag, Berlin, 1983.
  • [21] Y. Katnelson, An introduction to harmonic analysis, Wiley, New York, 1968.
  • [22] H. E. Lacey, The isometric theory of classical Banach spaces, Springer-Verlag, Berlin, 1974. MR 0493279 (58:12308)
  • [23] J. L. Rubio de Francia, Fourier series and Hilbert transforms with values in UMD Banach spaces, Studia Math. 81 (1985), 95-105. MR 818174 (87d:42008)
  • [24] J. L. Rubio de Francia and J. L. Torrea, Some Banach techniques in vector valued Fourier analysis, Colloq. Math. (to appear). MR 948520 (89g:46037)
  • [25] F. Ruiz and J. L. Torrea, Sobre el dual de espacios de Hardy de funciones con valores vectoriales, Proc. 8th Portuguese-Spanish Conf. Math., vol. I, Univ. of Coimbra, 1981, pp. 257-261.
  • [26] H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin, 1974. MR 0423039 (54:11023)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0951618-8
Keywords: Bounded mean oscillation measures, UMD spaces, Radon-Nikodym property, vector-valued atoms
Article copyright: © Copyright 1988 American Mathematical Society

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